Binomial distribution: formula, calculator and worked example

What the symbols mean

$P (X = k)$: the probability of exactly k successes
$P (X \leq k)$: the cumulative probability of at most k successes, $P(X ≤ k)$
$n$: the number of trials
$k$: the number of successes (0 to n)
$p$: the probability of success on each trial

When to use it

Use it when four conditions hold: a fixed number of trials n, each with two outcomes, the same success probability p every time, and trials independent of each other. Coin tosses, free throws, defective items in a batch and multiple-choice guessing all fit. For "at least k", use $1 − P(X ≤ k − 1)$ .

Worked example

A fair coin is tossed 10 times. What is the probability of exactly 3 heads? (n = 10, k = 3, p = 0.5)

Start from the formula.
$P (X = k) = (\binom{n}{k}) p^{k} \cdot {(1 - p)}^{n - k}$
Substitute $n$ = $10$ , $k$ = $3$ , $p$ = $0.5$ and work it out.
$P (X = k) = (\binom{10}{3}) \cdot {0.5}^{3} \cdot {(1 - 0.5)}^{10 - 3}$
$= 120 \cdot 0.125 \cdot {0.5}^{7}$
$= 15 \cdot 0.0078125$
$\approx 0.117188$
On average you would expect np = 5 successes in 10 trials.

Answer P(X = 3) = 0.1171875

Why it works

Any particular sequence with k successes and $n − k$ failures has probability $p^k(1 − p)^(n − k)$ , and there are $C(n, k)$ ways to choose which trials are the successes.

Common mistakes

Leaving out the $C(n, k)$ factor, which gives the probability of one particular order only.
Treating "at least 3" as "exactly 3"; use the cumulative probability and its complement.
Using it when trials are not independent, such as drawing cards without replacing them.

Binomial distribution

Calculator

See it move

What the symbols mean

When to use it

Worked example

Why it works

Common mistakes